Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:17:52Z http://mathoverflow.net/feeds/question/57851 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57851/are-all-shimura-varieties-special-subvarieties-of-the-siegel-modular-variety Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety? jacob 2011-03-08T16:31:03Z 2011-03-08T21:17:47Z <p>Given a Shimura variety \$S\$, is it possible to imbed \$S\$ as a special Subvariety of the Siegel modular variety \$A_{g,N}\$, for some \$g\$ and level \$N\$? I expect that the answer is yes, essentially since every semisimple group over \$\mathbb{Q}\$ should imbed into \$GL_n\$ via its adjoint representation, and \$GL_n\$ imbeds into \$SP_{2n}\$. However, I'm a bit worried about the business regarding weights.</p> <p>Thank you, Jacob</p> http://mathoverflow.net/questions/57851/are-all-shimura-varieties-special-subvarieties-of-the-siegel-modular-variety/57883#57883 Answer by Keerthi Madapusi Pera for Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety? Keerthi Madapusi Pera 2011-03-08T21:17:47Z 2011-03-08T21:17:47Z <p>The answer is no, in general. The problem is to find an embedding so that the minuscule character corresponding to the Shimura datum for \$S\$ induces the minuscule character of \$GSp_{2n}\$ corresponding to a decomposition into Lagrangians. </p> <p>In the affirmative direction, for most classical, simply connected groups (and only for classical groups, i.e. of types \$A\$,\$B\$,\$C\$ and \$D\$), the answer is yes; some subtleties crop up for \$Spin^*(2n)\$ (this is the so called \$D^{\mathbb{H}}\$ case), for which only the quotient by an order 2 central sub-group admits a symplectic embedding (of Shimura data).</p> <p>This is all beautifully laid out in Deligne's article 'Varietes de Shimura...' <a href="http://www.ams.org/mathscinet-getitem?mr=0546620" rel="nofollow">here</a>, following Satake <a href="http://www.jstor.org/stable/2373012" rel="nofollow">here</a>. See also Proposition 1.21 in Milne's article <a href="http://www.jmilne.org/math/articles/1994bP.pdf" rel="nofollow">here</a> </p>